Essentials Of Stochastic Processes 3rd Edition Richard Durrett - Solutions

A salesman flies around between Atlanta, Boston, and Chicago as the following rates (the units are trips per month):A B C A 4 2 2 B 3 4 1 C 5 0 5(a) Find the limiting fraction of time she spends in each city. (b) What is her average number of trips each year from Boston to Atlanta?
Suppose that the limiting age distribution in (3.10) is the same as the original distribution. Conclude that F.x/ D 1 ex for some > 0.
While visiting Haifa, Sid Resnick discovered that people who wish to travel from the port area up the mountain frequently take a shared taxi known as a sherut.The capacity of each car is five people. Potential customers arrive according to a Poisson process with rate . As soon as five people are
Each time the frozen yogurt machine at the mall breaks down, it is replaced by a new one of the same type. (a)What is the limiting age distribution for the machine in use if the lifetime of a machine has a gamma(2,) distribution, i.e., the sum of two exponentials with mean 1=. (b) Find the answer
The city of Ithaca, New York, allows for two-hour parking in all downtown spaces. Methodical parking officials patrol the downtown area, passing the same point every two hours. When an official encounters a car, he marks it with chalk. If the car is still there two hours later, a ticket is written.
Show that chain in Exercise 1.39 with transition probability isis a special case of the age chain. Use this observation and the previous exercise to compute the stationary distribution. 1 2 3 4 1 1/2 1/2 0 0 22/3 0 1/3 0 3 3/4 0 0 1/4 4 1 0 0 0
Consider the discrete renewal process with fj D P.t1 D j/ and Fi D P.t1 > i/.(a) Show that the age chain has transition probability(b) Show that if Et1 .i/ D P.t1 > i/=Et1. (c) Let p.i; j/ be the transition probability for the renewal chain. Verify that q is the dual chain of p, i.e., the
A scientist has a machine for measuring ozone in the atmosphere that is located in the mountains just north of Los Angeles. At times of a Poisson process with rate 1, storms or animals disturb the equipment so that it can no longer collect data. The scientist comes every L units of time to check
People arrive at a college admissions office at rate 1 per minute. When k people have arrive a tour starts. Student tour guides are paid $20 for each tour they conduct. The college estimates that it loses ten cents in good will for each minute a person waits. What tour group size minimizes the
A machine tool wears over time and may fail. The failure time measured in months has density fT .t/ D 2t=900 for 0 t 30 and 0 otherwise. If the tool fails, it must be replaced immediately at a cost of $1200. If it is replaced prior to failure, the cost is only $300. Consider a replacement
Consider the setup of Example 3.3 but now suppose that the car’s lifetime h.t/ D et. Show that for any A and B the optimal time T D 1. Can you give a simple verbal explanation?
Random Investment. An investor has $100,000. If the current interest rate is i% (compounded continuously so that the grow per year is exp.i=100/), he invests his money in a i year CD, takes the profits, and then reinvests the $100,000. Suppose that the kth investment leads to an interest rate Xk
In the Duke versusWake Forest football game, possessions alternate between Duke who has the ball for an average of 3.5 minutes and Wake Forest who has the ball for an average of 2.5 minutes. (a) In the long run what fraction of time does Duke have the ball? (b) Suppose that on each possession Duke
A worker has a number of machines to repair. Each time a repair is completed a new one is begun. Each repair independently takes an exponential amount of time with rate to complete. However, independent of this, mistakes occur according to a Poisson process with rate . Whenever a mistake occurs,
A young doctor is working at night in an emergency room. Emergencies come in at times of a Poisson process with rate 0.5 per hour. The doctor can only get to sleep when it has been 36 minutes (.6 hours) since the last emergency. For example, if there is an emergency at 1:00 and a second one at 1:17
One of the difficulties about probability is realizing when two different looking problems are the same, in this case dealing cocaine and fighting fires. In Problem 2.19, calls to a fire station arrive according to a Poisson process with rate 0.5 per hour, and the time required to respond to a
A cocaine dealer is standing on a street corner. Customers arrive at times of a Poisson process with rate . The customer and the dealer then disappear from the street for an amount of time with distribution G while the transaction is completed.Customers that arrive during this time go away never
Counter processes. Suppose that arrivals at a counter come at times of a Poisson process with rate . An arriving particle that finds the counter free gets registered and then locks the counter for an amount of time. Particles that arrive while the counter is locked have no effect. Find the
A policeman cruises (on average) approximately ten minutes before stopping a car for speeding. 90% of the cars stopped are given speeding tickets with an $80 fine. It takes the policeman an average of five minutes to write such a ticket. The other 10% of the stops are for more serious offenses,
Three children take turns shooting a ball at a basket. They each shoot until they miss and then it is next child’s turn. Suppose that child i makes a basket with probability pi and that successive trials are independent. (a) Determine the proportion of time in the long run that each child shoots.
In front of terminal C at the Chicago airport is an area where hotel shuttle vans park. Customers arrive at times of a Poisson process with rate 10 per hour looking for transportation to the Hilton hotel nearby. When seven people are in the van it leaves for the 36-minute round trip to the hotel.
The times between the arrivals of customers at a taxi stand are independent and have a distribution F with mean F. Assume an unlimited supply of cabs, such as might occur at an airport. Suppose that each customer pays a random fare with distribution G and mean G. Let W.t/ be the total fares paid up
Thousands of people are going to a Grateful dead concert in Pauley Pavillion at UCLA. They park their 10 foot cars on several of the long streets near the arena.There are no lines to tell the drivers where to park, so they park at random locations, and end up leaving spacings between the cars that
Monica works on a temporary basis. The mean length of each job she gets is 11 months. If the amount of time she spends between jobs is exponential with mean 3 months, then in the long run what fraction of the time does she spend working?
The weather in a certain locale consists of alternating wet and dry spells.Suppose that the number of days in each rainy spell is a Poisson distribution with mean 2, and that a dry spell follows a geometric distribution with mean 7. Assume that the successive durations of rainy and dry spells are
Consider two independent Poisson processes N1.t/ and N2.t/ with rates 1 and 2. What is the probability that the two-dimensional process .N1.t/;N2.t// ever visits the point .i; j/?
Starting at some fixed time, which we will call 0 for convenience, satellites are launched at times of a Poisson process with rate . After an independent amount of time having distribution function F and mean , the satellite stops working. Let X.t/ be the number of working satellites at time t.
Signals are transmitted according to a Poisson process with rate . Each signal is successfully transmitted with probability p and lost with probability 1 p.The fates of different signals are independent. For t 0 let N1.t/ be the number of signals successfully transmitted and let N2.t/ be the
Let t1; t2; : : : be independent exponential() random variables and let N be an independent random variable with P.N D n/ D .1 p/n1. What is the distribution of the random sum T D t1 C CtN?
Consider a Poisson process with rate and let L be the time of the last arrival in the interval OE0; t, with L D 0 if there was no arrival. (a) Compute E.t L/ (b)What happens when we let t!1in the answer to (a)?
Messages arrive to be transmitted across the internet at times of a Poisson process with rate . Let Yi be the size of the ith message, measured in bytes, and let g.z/ D EzYi be the generating function of Yi. Let N.t/ be the number of arrivals at time t and S D Y1 CCYN.t/ be the total size of the
Let fN.t/; t 0g be a Poisson process with rate . Let T 0 be an independent with mean and variance 2. Find cov.T;NT /.
Let St be the price of stock at time t and suppose that at times of a Poisson process with rate the price is multiplied by a random variable Xi > 0 with mean and variance 2. That is,where the product is 1 if N.t/ D 0. Find ES.t/ and var S.t/. N(t) S = Soxi i=1
Policy holders of an insurance company have accidents at times of a Poisson process with rate . The distribution of the time R until a claim is reported is random with P.R r/ D G.r/ and ER D . (a) Find the distribution of the number of unreported claims. (b) Suppose each claim has mean and
Consider a Poisson process with rate and let L be the time of the last arrival in the interval OE0; t, with L D 0 if there was no arrival. (a) Compute E.t L/ (b)What happens when we let t!1in the answer to (a)?
Let Ti, i D 1; 2; 3 be independent exponentials with rate i. (a) Show that for any numbers t1; t2; t3 maxft1; t2; t3g D t1 C t2 C t3 minft1; t2g minft1; t3g minft2; t3g C minft1; t2; t3g(b) Use (a) to find E maxfT1; T2; T3g. (c) Use the formula to give a simple solution of part (c) of
A flashlight needs two batteries to be operational. You start with n batteries numbered 1 to n. Whenever a battery fails it is replaced by the lowest-numbered working battery. Suppose that battery life is exponential with mean 100 hours. Let T be the time at which there is one working battery left
Consider a bank with two tellers. Three people, Alice, Betty, and Carol enter the bank at almost the same time and in that order. Alice and Betty go directly into service while Carol waits for the first available teller. Suppose that the service times for each teller are exponentially distributed
Let S and T be exponentially distributed with rates and . Let U D minfS; Tg, V D maxfS; Tg, and W D V U. Find the variances of U, V, and W.
Let S and T be exponentially distributed with rates and . Let U D minfS; Tg and V D maxfS; Tg. Find (a) EU. (b) E.V U/, (c) EV. (d) Use the identity V D SCT U to get a different looking formula for EV and verify the two are equal.
Customers arrive according to a Poisson process of rate per hour. Joe does not want to stay until the store closes at T D 10 p.m., so he decides to close up when the first customer after time T s arrives. He wants to leave early but he does not want to lose any business so he is happy if he
Copy machine 1 is in use now. Machine 2 will be turned on at time t. Suppose that the machines fail at rate i. What is the probability that machine 2 is the first to fail?
Ignoring the fact that the bar exam is only given twice a year, let us suppose that new lawyers arrive in Los Angeles according to a Poisson process with mean 300 per year. Suppose that each lawyer independently practices for an amount of time T with a distribution function F.t/ D P.T t/ that has
A policewoman on the evening shift writes a Poisson mean 6 number of tickets per hour. 2/3’s of these are for speeding and cost $100. 1/3’s of these are for DWI and cost $400. (a) Find the mean and standard deviation for the total revenue from the tickets she writes in an hour. (b) What is the
People arrive at the Southpoint Mall at times of a Poisson process with rate 96 per hour. 1/3 of the shoppers are men and 2/3 are women. (a)Women shop for an amount of time that is exponentially distributed with mean three hours. Men shop for a time that is uniformly distributed on OE0; 1 hour.
Vehicles carrying Occupy Durham protesters arrive at rate 30 per hour.• 50% are bicycles carrying one person• 30% are BMW’s carrying two people• 20% are Prius’s carrying four happy carpoolers(a) What is the probability exactly 4 Prius’s arrive between 12 and 12:30? (b) Find the mean and
People arrive at the Durham Farmer’s market at rate 15 per hour. 4/5’s are vegetarians, and 1/5 are meat eaters. Vegetarians spend an average of $7 with a standard deviation of 3. Meat eaters spend an average of $15 with a standard deviation of 8. (a) Compute the probability that in the first
A Philadelphia taxi driver gets new customers at rate 1/5 per minute. With probability 1/3 the person wants to go to the airport, a 20-minute trip. After waiting in a line of cabs at the airport for an average of 35 minutes, he gets another fare and spends 20 minutes driving back to drop that
Suppose that Virginia scores touchdowns at rate 7 and field goals at rate 3 while Duke scores touchdowns at rate 7 and field goals at rate 3. The subscripts indicate the number of points the team receives for each type of event. The final score in the Virginia-Duke game in 2015 was 42–34, i.e.,
As a community service members of the Mu Alpha Theta fraternity are going to pick up cans from along a roadway. A Poisson mean 60 members show up for work. 2/3 of the workers are enthusiastic and will pick up a mean of ten cans with a standard deviation of 5. 1/3 of the workers are lazy and will
Traffic on Snyder Hill Road in Ithaca, NY, follows a Poisson process with rate 2/3’s of a vehicle per minute. 10% of the vehicles are trucks, the other 90%are cars. (a) What is the probability at least one truck passes in a hour? (b) Given that ten trucks have passed by in an hour, what is the
Hockey teams 1 and 2 score goals at times of Poisson processes with rates 1 and 2. Suppose that N1.0/ D 3 and N2.0/ D 1. (a) What is the probability that N1.t/will reach 5 before N2.t/ does? (b) Answer part (a) for Poisson processes with rates1 and 2.
Wayne Gretsky scored a Poisson mean 6 number of points per game. 60%of these were goals and 40% were assists (each is worth one point). Suppose he is paid a bonus of 3K for a goal and 1K for an assist. (a) Find the mean and standard deviation for the total revenue he earns per game. (b) What is the
Customers arrive at a bank according to a Poisson process with rate 10 per hour. Given that two customers arrived in the first five minutes, what is the probability that (a) both arrived in the first two minutes. (b) at least one arrived in the first two minutes.
For a Poisson process N.t/ with arrival rate 2 compute: (a) P.N.2/ D 5/, (b)P.N.5/ D 8jN.2/ D 3, (c) P.N.2/ D 3jN.5/ D 8/.
Suppose N.t/ is a Poisson process with rate 2. Compute the conditional probabilities (a) P.N.3/ D 4jN.1/ D 1/, (b) P.N.1/ D 1jN.3/ D 4/.
Suppose that the number of calls per hour to an answering service follows a Poisson process with rate 4. Suppose that 3/4’s of the calls are made by men, 1/4 by women, and the sex of the caller is independent of the time of the call. (a) What is the probability that in one hour exactly two men
Calls originate from Dryden according to a rate 12 Poisson process. 3/4 are local and 1/4 are long distance. Local calls last an average of ten minutes, while long distance calls last an average of five minutes. Let M be the number of local calls and N the number of long distance calls in
A light bulb has a lifetime that is exponential with a mean of 200 days.When it burns out a janitor replaces it immediately. In addition there is a handyman who comes at times of a Poisson process at rate .01 and replaces the bulb as “preventive maintenance.” (a) How often is the bulb replaced?
Rock concert tickets are sold at a ticket counter. Females and males arrive at times of independent Poisson processes with rates 30 and 20 customers per hour.(a) What is the probability the first three customers are female? (b) If exactly two customers arrived in the first five minutes, what is the
Customers arrive at an automated teller machine at the times of a Poisson process with rate of 10 per hour. Suppose that the amount of money withdrawn on each transaction has a mean of $30 and a standard deviation of $20. Find the mean and standard deviation of the total withdrawals in eight hours.
An insurance company pays out claims at times of a Poisson process with rate 4 per week. Writing K as shorthand for “thousands of dollars,” suppose that the mean payment is 10K and the standard deviation is 6K. Find the mean and standard deviation of the total payments for 4 weeks.
Edwin catches trout at times of a Poisson process with rate 3 per hour.Suppose that the trout weigh an average of four pounds with a standard deviation of two pounds. Find the mean and standard deviation of the total weight of fish he catches in two hours.
When did the chicken cross the road? Suppose that traffic on a road follows a Poisson process with rate cars per minute. A chicken needs a gap of length at least c minutes in the traffic to cross the road. To compute the time the chicken will have to wait to cross the road, let t1; t2; t3; : : :
Let T be exponentially distributed with rate . (a) Use the definition of conditional expectation to compute E.TjT < c/. (b) Determine E.TjT < c/ from the identity ET D P.T < c/E.TjT < c/ C P.T > c/E.TjT > c/
The number of hours between successive trains is T which is uniformly distributed between 1 and 2. Passengers arrive at the station according to a Poisson process with rate 24 per hour. Let X denote the number of people who get on a train.Find (a) EX, (b) var .X/.
A math professor waits at the bus stop at the Mittag-Leffler Institute in the suburbs of Stockholm, Sweden. Since he has forgotten to find out about the bus schedule, his waiting time until the next bus is uniform on (0,1). Cars drive by the bus stop at rate 6 per hour. Each will take him into town
Calls to the Dryden fire department arrive according to a Poisson process with rate 0.5 per hour. Suppose that the time required to respond to a call, return to the station, and get ready to respond to the next call is uniformly distributed between 1/2 and 1 hour. If a new call comes before the
Traffic on Rosedale Road in Princeton, NJ, follows a Poisson process with rate 6 cars per minute. A deer runs out of the woods and tries to cross the road. If there is a car passing in the next five seconds, then there will be a collision. (a) Find the probability of a collision. (b) What is the
Suppose that the number of calls per hour to an answering service follows a Poisson process with rate 4. (a) What is the probability that fewer (i.e.,
Customers arrive at a shipping office at times of a Poisson process with rate 3 per hour. (a) The office was supposed to open at 8 a.m. but the clerk Oscar overslept and came in at 10 a.m. What is the probability that no customers came in the twohour period? (b) What is the distribution of the
Suppose N.t/ is a Poisson process with rate 3. Let Tn denote the time of the nth arrival. Find (a) E.T12/, (b) E.T12jN.2/ D 5/, (c) E.N.5/jN.2/ D 5/.
Suppose 1% of a certain brand of Christmas lights is defective. Use the Poisson approximation to compute the probability that in a box of 25 there will be at most one defective bulb.
The probability of a three of a kind in poker is approximately 1/50. Use the Poisson approximation to estimate the probability you will get at least one three of a kind if you play 20 hands of poker.
Compare the Poisson approximation with the exact binomial probabilities of no success when (a) n D 10, p D 0:1, (b) n D 50, p D 0:02.
Compare the Poisson approximation with the exact binomial probabilities of 1 success when n D 20, p D 0:1.
Ron, Sue, and Ted arrive at the beginning of a professor’s office hours. The amount of time they will stay is exponentially distributed with means of 1, 1/2, and 1/3 hour. (a) What is the expected time until only one student remains? (b) For each student find the probability they are the last
Excited by the recent warm weather Jill and Kelly are doing spring cleaning at their apartment. Jill takes an exponentially distributed amount of time with mean 30 minutes to clean the kitchen. Kelly takes an exponentially distributed amount of time with mean 40 minutes to clean the bathroom. The
A flashlight needs two batteries to be operational. You start with four batteries numbered 1–4. Whenever a battery fails it is replaced by the lowest-numbered working battery. Suppose that battery life is exponential with mean 100 hours. Let T be the time at which there is one working battery
Consider a bank with two tellers. Three people, Anne, Betty, and Carol enter the bank at almost the same time and in that order. Anne and Betty go directly into service while Carol waits for the first available teller. Suppose that the service times for two servers are exponentially distributed
In a hardware store you must first go to server 1 to get your goods and then go to a server 2 to pay for them. Suppose that the times for the two activities are exponentially distributed with means six and three minutes. Compute the average amount of time it takes Bob to get his goods and pay if
Ilan and Justin are competing in a math competition. They work independently and each has the same two problems to solve. The two problems take an exponentially distributed amount of time with mean 20 and 30 minutes respectively (or rates 3 and 2 if written in terms of hours). (a) What is the
Three people are fishing and each catches fish at rate 2 per hour. How long do we have to wait until everyone has caught at least one fish?
A doctor has appointments at 9 and 9:30. The amount of time each appointment lasts is exponential with mean 30. What is the expected amount of time after 9:30 until the second patient has completed his appointment?
The lifetime of a radio is exponentially distributed with mean 5 years. If Ted buys a 7-year-old radio, what is the probability it will be working 3 years later?
Suppose that the time to repair a machine is exponentially distributed random variable with mean 2. (a) What is the probability the repair takes more than two hours. (b) What is the probability that the repair takes more than five hours given that it takes more than three hours.
Consider a branching process as defined in 1.55, in which each family has a number of children that follows a shifted geometric distribution: pk D p.1 p/k for k 0, which counts the number of failures before the first success when success has probability p. Compute the probability that starting
Consider a branching process as defined in 1.55, in which each family has exactly three children, but invert Galton and Watson’s original motivation and ignore male children. In this model a mother will have an average of 1.5 daughters.Compute the probability that a given woman’s descendents
The opposite of the aging chain is the renewal chain with state space f0; 1; 2; : : :g in which p.i; i 1/ D 1 when i > 0. The only nontrivial part of the transition probability is p.0; i/ D pi. Show that this chain is always recurrent but is positive recurrent if and only ifPn npn < 1.
Consider the aging chain on f0; 1; 2; : : :g in which for any n 0 the individual gets one day older from n to nC1 with probability pn but dies and returns to age 0 with probability 1pn. Find conditions that guarantee that (a) 0 is recurrent,(b) positive recurrent. (c) Find the stationary
Consider the Markov chain with state space f1;2; : : :g and transition probabilityand p.1; 1/ D 1 p.1; 2/ D 3=4. Show that there is no stationary distribution. p(m, m+1)=m/(2m + 2) for m 1 p(m,m-1) = 1/2 for m 2 p(m,m)=1/(2m + 2) for m 2
Consider the Markov chain with state space f0; 1; 2; : : :g and transition probabilityand p.0; 0/ D 1 p.0; 1/ D 3=4. Find the stationary distribution . p(m, m + 1) = p(m,m-1) 1/2(1-m+2) 1/(1+m 1 m+2 for m 0 for m 1
To see what the conditions in the last problem say we will now consider some concrete examples. Let px D 1=2, qx D ecx˛=2, rx D 1=2 qx for x 1 and p0 D 1. For large x, qx .1 cx˛/=2, but the exponential formulation keeps the probabilities nonnegative and makes the problem easier to solve.
General birth and death chains. The state space is f0; 1; 2; : : :g and the transition probability has p.x; x C 1/ D px p.x; x 1/ D qx for x > 0 p.x; x/ D rx for x 0 while the other p.x; y/ D 0. Let Vy D minfn 0 W Xn D yg be the time of the first visit to y and let hN.x/ D Px.VN < V0/. By
Algorithmic efficiency. The simplex method minimizes linear functions by moving between extreme points of a polyhedral region so that each transition decreases the objective function. Suppose there are n extreme points and they are numbered in increasing order of their values. Consider the Markov
Coupon collector’s problem. We are interested now in the time it takes to collect a set of N baseball cards. Let Tk be the number of cards we have to buy before we have k that are distinct. Clearly, T1 D 1. A little more thought reveals that if each time we get a card chosen at random from all N
Roll a fair die repeatedly and let Y1; Y2; : : : be the resulting numbers. Let Xn D jfY1; Y2; : : : ; Yngj be the number of values we have seen in the first n rolls for n 1 and set X0 D 0. Xn is a Markov chain. (a) Find its transition probability.(b) Let T D minfn W Xn D 6g be the number of
Brother–sister mating. In this genetics scheme two individuals (one male and one female) are retained from each generation and are mated to give the next. If the individuals involved are diploid and we are interested in a trait with two alleles, A anda, then each individual has three possible
Ehrenfest chain. Consider the Ehrenfest chain, Example 1.2, with transition probability p.i; i C 1/ D .N i/=N, and p.i; i 1/ D i=N for 0 i N. Let n D ExXn. (a) Show that nC1 D 1 C .1 2=N/ n. (b) Use this and induction to conclude thatFrom this we see that the mean n converges exponentially
Wright–Fisher model. Consider the chain described in Example 1.7.where x D .1 u/x=N C v.N x/=N. (a) Show that if u; v > 0, then limn!1 pn.x; y/ D .y/, where is the unique stationary distribution. There is no known formula for .y/, but you can (b) compute the mean D Py y.y/ D limn!1 ExXn.

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Essentials Of Stochastic Processes 3rd Edition Richard Durrett - Solutions

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